Integrand size = 19, antiderivative size = 26 \[ \int \cot ^2(c+d x) (a+b \sec (c+d x)) \, dx=-a x-\frac {\cot (c+d x) (a+b \sec (c+d x))}{d} \]
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Time = 0.03 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {3967, 8} \[ \int \cot ^2(c+d x) (a+b \sec (c+d x)) \, dx=-\frac {\cot (c+d x) (a+b \sec (c+d x))}{d}-a x \]
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Rule 8
Rule 3967
Rubi steps \begin{align*} \text {integral}& = -\frac {\cot (c+d x) (a+b \sec (c+d x))}{d}-\int a \, dx \\ & = -a x-\frac {\cot (c+d x) (a+b \sec (c+d x))}{d} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.01 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.65 \[ \int \cot ^2(c+d x) (a+b \sec (c+d x)) \, dx=-\frac {b \csc (c+d x)}{d}-\frac {a \cot (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},-\tan ^2(c+d x)\right )}{d} \]
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Time = 0.46 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.35
method | result | size |
derivativedivides | \(\frac {a \left (-\cot \left (d x +c \right )-d x -c \right )-\frac {b}{\sin \left (d x +c \right )}}{d}\) | \(35\) |
default | \(\frac {a \left (-\cot \left (d x +c \right )-d x -c \right )-\frac {b}{\sin \left (d x +c \right )}}{d}\) | \(35\) |
risch | \(-a x -\frac {2 i \left (b \,{\mathrm e}^{i \left (d x +c \right )}+a \right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}\) | \(38\) |
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none
Time = 0.27 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.27 \[ \int \cot ^2(c+d x) (a+b \sec (c+d x)) \, dx=-\frac {a d x \sin \left (d x + c\right ) + a \cos \left (d x + c\right ) + b}{d \sin \left (d x + c\right )} \]
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\[ \int \cot ^2(c+d x) (a+b \sec (c+d x)) \, dx=\int \left (a + b \sec {\left (c + d x \right )}\right ) \cot ^{2}{\left (c + d x \right )}\, dx \]
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none
Time = 0.28 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.19 \[ \int \cot ^2(c+d x) (a+b \sec (c+d x)) \, dx=-\frac {{\left (d x + c + \frac {1}{\tan \left (d x + c\right )}\right )} a + \frac {b}{\sin \left (d x + c\right )}}{d} \]
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none
Time = 0.30 (sec) , antiderivative size = 52, normalized size of antiderivative = 2.00 \[ \int \cot ^2(c+d x) (a+b \sec (c+d x)) \, dx=-\frac {2 \, {\left (d x + c\right )} a - a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {a + b}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}}{2 \, d} \]
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Time = 14.34 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.85 \[ \int \cot ^2(c+d x) (a+b \sec (c+d x)) \, dx=\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {a}{2}-\frac {b}{2}\right )}{d}-\frac {\frac {a}{2}+\frac {b}{2}}{d\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}-a\,x \]
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